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Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009, Article ID 762478, 17 pages doi:10.1155/2009/762478 ResearchArticleGlobalAttractivityResultsforMixed-MonotoneMappingsinPartiallyOrderedCompleteMetric Spaces D ˇ z. Burgi ´ c, 1 S. Kalabu ˇ si ´ c, 2 andM.R.S.Kulenovi ´ c 3 1 Department of Mathematics, University of Tuzla, 75000 Tuzla, Bosnia and Herzegovina 2 Department of Mathematics, University of Sarajevo, 71000 Sarajevo, Bosnia and Herzegovina 3 Department of Mathematics, University of Rhode Island, Kingston, R I 02881-0816, USA Correspondence should be addressed to M. R. S. Kulenovi ´ c, mkulenovic@mail.uri.edu Received 28 October 2008; Revised 17 January 2009; Accepted 9 February 2009 Recommended by Juan J. Nieto We prove fixed point theorems formixed-monotonemappingsinpartiallyorderedcompletemetric spaces which satisfy a weaker contraction condition than the classical Banach contraction condition for all points that are related by given ordering. We also give a globalattractivity result for all solutions of the difference equation z n1 Fz n ,z n−1 , n 2, 3, ,where F satisfies mixed- monotone conditions with respect to the given ordering. Copyright q 2009 D ˇ z. Burgi ´ c et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction and Preliminaries The following results were obtained first in 1 and were extended to the case of higher- order difference equations and systems in 2–6. For the sake of completeness and the readers convenience, we are including short proofs. Theorem 1.1. Let a, b be a compact interval of real numbers, and assume that f : a, b × a, b −→ a, b1.1 is a continuous function satisfying the following properties: afx, y is nondecreasing in x ∈ a, b for each y ∈ a, b, and fx, y is nonincreasing in y ∈ a, b for each x ∈ a, b; 2 Fixed Point Theory and Applications b If m, M ∈ a, b × a, b is a solution of the system fm, Mm, fM, mM, 1.2 then m M. Then x n1 f x n ,x n−1 ,n 0, 1, 1.3 has a unique equilibrium x ∈ a, b and every solution of 1.3 converges to x. Proof. Set m 0 a, M 0 b, 1.4 and for i 1, 2, set M i f M i−1 ,m i−1 ,m i f m i−1 ,M i−1 . 1.5 Now observe that for each i ≥ 0, m 0 ≤ m 1 ≤··· ≤ m i ≤··· ≤ M i ≤··· ≤ M 1 ≤ M 0 , m i ≤ x k ≤ M i , for k ≥ 2i 1. 1.6 Set m lim i →∞ m i ,M lim i →∞ M i . 1.7 Then M ≥ lim sup i →∞ x i ≥ lim inf i →∞ x i ≥ m 1.8 and by the continuity of f, m fm, M,M fM, m. 1.9 Therefore in view of b, m M 1.10 from which the result follows. D ˇ z. Burgi ´ cetal. 3 Theorem 1.2. Let a, b be an interval of real numbers and assume that f : a, b × a, b −→ a, b1.11 is a continuous function satisfying the following properties: a fx, y is nonincreasing in x ∈ a, b for each y ∈ a, b, and fx, y is nondecreasing in y ∈ a, b for each x ∈ a, b; b the difference equation 1.3 has no solutions of minimal period two in a, b.Then1.3 has a unique equilibrium x ∈ a, b and every solution of 1.3 converges to x. Proof. Set m 0 a, M 0 b 1.12 and for i 1, 2, set M i f m i−1 ,M i−1 ,m i f M i−1 ,m i−1 . 1.13 Now observe that for each i ≥ 0, m 0 ≤ m 1 ≤··· ≤ m i ≤··· ≤ M i ≤··· ≤ M 1 ≤ M 0 , m i ≤ x k ≤ M i , for k ≥ 2i 1. 1.14 Set m lim i →∞ m i ,M lim i →∞ M i . 1.15 Then clearly 1.8 holds and by the continuity of f, m fM, m,M fm, M. 1.16 In view of b, m M 1.17 from which the result follows. These results have been very useful in proving attractivityresultsfor equilibrium or periodic solutions of 1.3 as well as for higher-order difference equations and systems of difference equations; see 2, 7–12. Theorems 1.1 and 1.2 have attracted considerable attention of the leading specialists in difference equations and discrete dynamical systems and have been generalized and extended to the case of maps in R n ,see3, and maps in Banach space 4 Fixed Point Theory and Applications with the cone see 4–6. In this paper, we will extend Theorems 1.1 and 1.2 to the case of monotone mappingsinpartiallyorderedcompletemetric spaces. On the other hand, there has been recent interest in establishing fixed point theorems inpartiallyorderedcompletemetric spaces with a contractivity condition which holds for all points that are related by partial ordering; see 13–20. These fixed point results have been applied mainly to the existence of solutions of boundary value problems for differential equations and one of them, namely 20, has been applied to the problem of solving matrix equations. See also 21, where the application to the boundary value problems for integro- differential equations is given and 22 for application to some classes of nonexpansive mappings and 23 for the application of the Leray-Schauder theory to the problems of an impulsive boundary value problem under the condition of non-well-ordered upper and lower solutions. None of these results is global result, but they are rather existence results. In this paper, we combine the existence results with the results of the type of Theorems 1.1 and 1.2 to obtain globalattractivity results. 2. Main Results: Mixed Monotone Case I Let X be a partiallyordered set and let d be a metric on X such that X, d is a completemetric space. Consider X × X. We will use the following partial ordering. For x, y, u, v ∈ X × X, we have x, y u, v ⇐⇒ { x ≤ u, y ≥ v}. 2.1 This partial ordering is well known as “south-east ordering” in competitive systems in the plane; see 5, 6, 12, 24, 25. Let d 1 be a metric on X × X defined as follows: d 1 x, y, u, v dx, udy, v. 2.2 Clearly d 1 x, y, u, v d 1 y, x, v, u. 2.3 We prove the following theorem. Theorem 2.1. Let F : X × X → X be a map such that Fx, y is nonincreasing in x for all y ∈ X, and nondecreasing in y for all x ∈ X. Suppose that the following conditions hold. i There exists k ∈ 0, 1 with dFx, y,Fu, v ≤ k 2 d 1 x, y, u, v ∀x, y u, v. 2.4 ii There exists x 0 ,y 0 ∈ X such that the following condition holds: x 0 ≤ F y 0 ,x 0 ,y 0 ≥ F x 0 ,y 0 . 2.5 D ˇ z. Burgi ´ cetal. 5 iii If {x n }∈X is a nondecreasing convergent sequence such that lim n →∞ x n x,then x n ≤ x, for all n ∈ N and if {y n }∈Y is a nonincreasing convergent sequence such that lim n →∞ y n y,theny n ≥ y, for all n ∈ N;ifx n ≤ y n for every n,then lim n →∞ x n ≤ lim n →∞ y n . Then we have the following. a For every initial point x 0 ,y 0 ∈ X × X such that condition 2.5 holds, F n x 0 ,y 0 → x, F n y 0 ,x 0 → y, n →∞,wherex, y satisfy x Fy, x,y Fx, y. 2.6 If x 0 ≤ y 0 in condition 2.5,thenx ≤ y. If in addition x y,then{x n }, {y n } converge to the equilibrium of the equation x n1 F y n ,x n ,y n1 F x n ,y n ,n 1, 2, 2.7 b In particular, every solution {z n } of z n1 F z n ,z n−1 ,n 2, 3, 2.8 such that x 0 ≤ z 0 ,z 1 ≤ y 0 converges to the equilibrium of 2.8. c The following estimates hold: d F n y 0 ,x 0 ,x ≤ 1 2 k n 1 − k d F x 0 ,y 0 ,y 0 d F y 0 ,x 0 ,x 0 , 2.9 d F n x 0 ,y 0 ,y ≤ 1 2 k n 1 − k d F y 0 ,x 0 ,x 0 d F x 0 ,y 0 ,y 0 . 2.10 Proof. Let x 1 Fy 0 ,x 0 and y 1 Fx 0 ,y 0 . Since x 0 ≤ Fy 0 ,x 0 x 1 and y 0 ≥ Fx 0 ,y 0 y 1 , for x 2 Fy 1 ,x 1 ,y 2 Fx 1 ,y 1 , we have F 2 y 0 ,x 0 : F F x 0 ,y 0 ,F y 0 ,x 0 F y 1 ,x 1 x 2 , F 2 x 0 ,y 0 : F F y 0 ,x 0 ,F x 0 ,y 0 F x 1 ,y 1 y 2 . 2.11 Now, we have x 2 F 2 y 0 ,x 0 F y 1 ,x 1 ≥ F y 0 ,x 0 x 1 , y 2 F 2 x 0 ,y 0 F x 1 ,y 1 ≤ F x 0 ,y 0 y 1 . 2.12 6 Fixed Point Theory and Applications For n 1, 2, ,we let x n1 F n1 y 0 ,x 0 F F n x 0 ,y 0 ,F n y 0 ,x 0 , y n1 F n1 x 0 ,y 0 F F n y 0 ,x 0 ,F n x 0 ,y 0 . 2.13 By using the monotonicity of F,weobtain x 0 ≤ F y 0 ,x 0 x 1 ≤ F 2 y 0 ,x 0 x 2 ≤··· ≤ F n1 y 0 ,x 0 ≤··· , y 0 ≥ F x 0 ,y 0 y 1 ≥ F 2 x 0 ,y 0 y 2 ≥··· ≥ F n1 x 0 ,y 0 ≥··· 2.14 that is x 0 ≤ x 1 ≤ x 2 ≤··· y 0 ≥ y 1 ≥ y 2 ≥··· . 2.15 We claim that for all n ∈ N the following inequalities hold: d x n1 ,x n d F n1 y 0 ,x 0 ,F n y 0 ,x 0 ≤ k n 2 d 1 x 1 ,y 1 , x 0 ,y 0 , 2.16 d y n1 ,y n d F n1 x 0 ,y 0 ,F n x 0 ,y 0 ≤ k n 2 d 1 x 1 ,y 1 , x 0 ,y 0 . 2.17 Indeed, for n 1, using x 0 ≤ Fy 0 ,x 0 , y 0 ≥ Fx 0 ,y 0 ,and2.3,weobtain d x 2 ,x 1 d F y 1 ,x 1 ,F y 0 ,x 0 ≤ k 2 d 1 y 1 ,x 1 , y 0 ,x 0 k 2 d 1 x 1 ,y 1 , x 0 ,y 0 , d y 2 ,y 1 d Fx 1 ,y 1 ,F x 0 ,y 0 ≤ k 2 d 1 x 1 ,y 1 , x 0 ,y 0 . 2.18 Assume that 2.16 holds. Using the inequalities F n1 y 0 ,x 0 ≥ F n y 0 ,x 0 , F n1 x 0 ,y 0 ≤ F n x 0 ,y 0 , 2.19 D ˇ z. Burgi ´ cetal. 7 and the contraction condition 2.4, we have d x n2 ,x n1 d F n2 y 0 ,x 0 ,F n1 y 0 ,x 0 d F F n1 x 0 ,y 0 ,F n1 y 0 ,x 0 ,F F n x 0 ,y 0 ,F n y 0 ,x 0 ≤ k 2 d F n1 x 0 ,y 0 ,F n x 0 ,y 0 d F n1 y 0 ,x 0 ,F n y 0 ,x 0 ≤ k 2 k n 2 d F x 0 ,y 0 ,y 0 d F y 0 ,x 0 ,x 0 d F y 0 ,x 0 ,x 0 d F x 0 ,y 0 ,y 0 k n1 2 d 1 x 1 ,y 1 , x 0 ,y 0 . 2.20 Similarly, d y n2 ,y n1 d F n2 x 0 ,y 0 ,F n1 x 0 ,y 0 ≤ k n1 2 d 1 x 1 ,y 1 , x 0 ,y 0 . 2.21 This implies that {x n } {F n y 0 ,x 0 } and {y n } {F n x 0 ,y 0 } are Cauchy sequences in X. Indeed, d F n y 0 ,x 0 ,F np y 0 ,x 0 ≤ d F n y 0 ,x 0 ,F n1 y 0 ,x 0 ··· d F np−1 y 0 ,x 0 ,F np y 0 ,x 0 ≤ k n 2 d F x 0 ,y 0 ,y 0 d F y 0 ,x 0 ,x 0 ··· k np−1 2 d F x 0 y 0 ,y 0 d F y 0 ,x 0 ,x 0 k n 2 1 k k 2 ··· k p−1 d F x 0 ,y 0 ,y 0 dF y 0 ,x 0 ,x 0 k n 2 1 − k p 1 − k d F x 0 ,y 0 ,y 0 d F y 0 ,x 0 ,x 0 . 2.22 Since k ∈ 0, 1, we have d x n ,x np d F n y 0 ,x 0 ,F np y 0 ,x 0 ≤ k n 21 − k d 1 x 1 ,y 1 , x 0 ,y 0 . 2.23 8 Fixed Point Theory and Applications Using 2.23, we conclude that {x n } {F n y 0 ,x 0 } is a Cauchy sequence. Similarly, we conclude that {y n } {F n x 0 ,y 0 } is a Cauchy sequence. Since X is a completemetric space, then there exist x, y ∈ X such that lim n →∞ x n lim n →∞ F n y 0 ,x 0 x, lim n →∞ y n lim m →∞ F m x 0 ,y 0 y. 2.24 Using the continuity of F, which follows from contraction condition 2.4, the equations x n1 F y n ,x n ,y n1 F x n ,y n 2.25 imply 2.6. Assume that x 0 ≤ y 0 . Then, in view of the monotonicity of F x 1 F y 0 ,x 0 ≤ F x 0 ,y 0 y 1 , x 2 F y 1 ,x 1 ≤ F x 1 ,y 1 y 2 , x 3 F y 2 ,x 2 ≤ F x 2 ,y 2 y 3 . 2.26 By using induction, we can show that x n ≤ y n for all n. Assume that x 0 ≤ z 0 ,z 1 ≤ y 0 . Then, in view of the monotonicity of F, we have x 1 F y 0 ,x 0 ≤ F z 1 ,z 0 z 2 ≤ F x 0 ,y 0 y 1 , x 1 F y 0 ,x 0 ≤ F z 2 ,z 1 z 3 ≤ F x 0 ,y 0 y 1 . 2.27 Continuing in a similar way we can prove that x i ≤ z k ≤ y i for all k ≥ 2i1. By using condition iii we conclude that whenever lim n →∞ z k exists we must have x ≤ lim k →∞ z k ≤ y 2.28 which in the case when x y implies lim k →∞ z k x. By letting p →∞in 2.23, we obtain the estimate 2.9. Remark 2.2. Property iii is usually called closedness of the partial ordering, see 6,andis an important ingredient of the definition of an ordered L-space; see 17, 19. Theorem 2.3. Assume that along with conditions (i) and (ii) of Theorem 2.1, the following condition is satisfied: iv every pair of elements has either a lower or an upper bound. Then, the fixed point x, y is unique and x y. Proof. First, we prove that the fi xed point x, y is unique. Condition iv is equivalent to the following. For every x, y, x ∗ ,y ∗ ∈ X × X, there exists z 1 ,z 2 ∈ X × X that is comparable to x, y, x ∗ ,y ∗ . See 16. Let x, y and x ∗ ,y ∗ be two fixed points of the map F. D ˇ z. Burgi ´ cetal. 9 We consider two cases. Case 1. If x, y is comparable to x ∗ ,y ∗ , then for all n 0, 1, 2, F n y, x,F n x, y is comparable to F n y ∗ ,x ∗ ,F n x ∗ ,y ∗ x ∗ ,y ∗ . We have to prove that d 1 x, y, x ∗ ,y ∗ 0. 2.29 Indeed, using 2.2,weobtain d 1 x, y, x ∗ ,y ∗ d x, x ∗ d y, y ∗ d F n y, x,F n y ∗ ,x ∗ d F n x, y,F n x ∗ ,y ∗ . 2.30 We estimate dF n y, x,F n y ∗ ,x ∗ ,anddF n x, y,F n x ∗ ,y ∗ . First, by using contraction condition 2.4, we have d Fy, x,F y ∗ ,x ∗ ≤ k 2 d y, y ∗ d x, x ∗ k 2 d 1 x, y, x ∗ ,y ∗ , d Fx, y,F x ∗ ,y ∗ ≤ k 2 d x, x ∗ d y, y ∗ k 2 d 1 x, y, x ∗ ,y ∗ . 2.31 Now, by using 2.31 and 2.30, we have d 1 x, y, x ∗ ,y ∗ ≤ kd 1 x, y, x ∗ ,y ∗ <d 1 x, y, x ∗ ,y ∗ , 2.32 which implies that d 1 x, y, x ∗ ,y ∗ 0. 2.33 Case 2. If x, y is not comparable to x ∗ ,y ∗ , then there exists an upper bound or a lower bound z 1 ,z 2 of x, y and x ∗ ,y ∗ . Then, F n z 2 ,z 1 ,F n z 1 ,z 2 is comparable to F n y, x,F n x, y and F n y ∗ ,x ∗ ,F n x ∗ ,y ∗ . Therefore, we have d 1 x, y, x ∗ ,y ∗ d 1 F n y, x,F n x, y , F n y ∗ ,x ∗ ,F n x ∗ ,y ∗ ≤ d 1 F n y, x,F n x, y , F n z 2 ,z 1 ,F n z 1 ,z 2 d 1 F n z 2 ,z 1 ,F n z 1 ,z 2 , F n y ∗ ,x ∗ ,F n x ∗ ,y ∗ d F n y, x,F n z 2 ,z 1 d F n z 2 ,z 1 ,F n y ∗ ,x ∗ d F n z 1 ,z 2 ,F n x ∗ ,y ∗ d F n z 2 ,z 1 ,F n y ∗ ,x ∗ . 2.34 10 Fixed Point Theory and Applications Now, we obtain d 1 x, y, x ∗ ,y ∗ d F n y, x,F n z 2 ,z 1 d F n z 2 ,z 1 ,F n y ∗ ,x ∗ d F n z 1 ,z 2 ,F n x ∗ ,y ∗ d F n z 2 ,z 1 ,F n y ∗ ,x ∗ . 2.35 We now estimate the right-hand side of 2.35. First, by using d Fy, x,F z 2 ,z 1 ≤ k 2 d y, z 2 d x, z 1 , 2.36 we have d F 2 y, x,F 2 z 2 ,z 1 d FFx, y,Fy,x,F F z 1 ,z 2 ,F z 2 ,z 1 ≤ k 2 d Fx, y,F z 1 ,z 2 d Fy, x,F z 2 ,z 1 ≤ k 2 k 2 d x, z 1 d y, z 2 k 2 d y, z 2 d x, z 1 k 2 2 d x, z 1 dy, z 2 . 2.37 Similarly, d F 2 x, y,F 2 z 1 ,z 2 d FFy,x,Fx, y,F F z 2 ,z 1 ,F z 1 ,z 2 ≤ k 2 d Fy, x,F z 2 ,z 1 d Fx, y ,F z 1 ,z 2 ≤ k 2 k 2 d y, z 2 d x, z 1 k 2 d y, z 2 d x, z 1 k 2 2 d x, z 1 d y, z 2 . 2.38 So, d F 2 y, x,F 2 z 2 ,z 1 ≤ k 2 2 d x, z 1 d y, z 2 , d F 2 x, y,F 2 z 1 ,z 2 ≤ k 2 2 d x, z 1 d y, z 2 . 2.39 [...]... proofs will be skipped Significant parts of these results have been included in 14 and applied successfully to some boundary value problems in ordinary differential equations Dˇ Burgi´ et al z c 15 Theorem 3.1 Let F : X × X → X be a map such that F x, y is nondecreasing in x for all y ∈ X, and nonincreasing in y for all x ∈ X Suppose that the following conditions hold i There exists k ∈ 0, 1 with ≤... “Fixed point theorems for generalized contractions inorderedmetric spaces,” Journal of Mathematical Analysis and Applications, vol 341, no 2, pp 1241–1252, 2008 19 A Petrusel and I A Rus, “Fixed point theorems inordered L-spaces,” Proceedings of the American ¸ Mathematical Society, vol 134, no 2, pp 411–418, 2006 Dˇ Burgi´ et al z c 17 20 A C M Ran and M C B Reurings, “A fixed point theorem in partially. .. contractions inpartiallyorderedmetric spaces,” Applicable Analysis, vol 87, no 1, pp 109–116, 2008 14 T Gnana Bhaskar and V Lakshmikantham, “Fixed point theorems inpartiallyorderedmetric spaces and applications,” Nonlinear Analysis: Theory, Methods & Applications, vol 65, no 7, pp 1379–1393, 2006 15 J J Nieto and R Rodr´guez-Lopez, “Existence and uniqueness of fixed point inpartiallyordered ı ´ sets... y 2.60 So, d x, y 2.61 3 Main Results: Mixed Monotone Case II Let X be a partiallyordered set and let d be a metric on X such that X, d is a completemetric space Consider X × X We will use the following partial order For x, y , u, v ∈ X × X, we have x, y u, v ⇐⇒ {x ≥ u, y ≤ v} 3.1 Let d1 be a metric on X × X defined as follows: d1 x, y , u, v d x, u d y, v 3.2 The following two theorems have similar... applications to ordinary differential equations,” Acta Mathematica Sinica, vol 23, no 12, pp 2205–2212, 2007 16 J J Nieto and R Rodr´guez-Lopez, “Contractive mapping theorems inpartiallyordered sets and ı ´ applications to ordinary differential equations,” Order, vol 22, no 3, pp 223–239, 2005 17 J J Nieto, R L Pouso, and R Rodr´guez-Lopez, “Fixed point theorems inordered abstract spaces,” ı ´ Proceedings of... the following condition holds: x0 ≤ F x0 , y0 , y0 ≥ F y0 , x0 3.4 x, then iii If {xn } ∈ X is a nondecreasing convergent sequence such that limn → ∞ xn xn ≤ x, for all n ∈ N and if {yn } ∈ Y is a nonincreasing convergent sequence such y, then yn ≥ y, for all n ∈ N; if xn ≤ yn for every n, then that limn → ∞ yn limn → ∞ xn ≤ limn → ∞ yn Then we have the following a For every initial point x0 , y0... following condition is satisfied: iv every pair of elements has either a lower or an upper bound Then, the fixed point x, y is unique and x y 16 Fixed Point Theory and Applications Remark 3.3 Theorems 3.1 and 3.2 generalize and extend the resultsin 14 The new feature of our results is globalattractivity part that extends Theorems 1.1 and 1.2 Most of presented ideas were presented for the first time in. .. of monotone hybrid algorithm for hemi-relatively nonexpansive mappings, ” Fixed Point Theory and Applications, vol 2008, Article ID 284613, 8 pages, 2008 23 X Xian, D O’Regan, and R P Agarwal, “Multiplicity results via topological degree for impulsive boundary value problems under non-well -ordered upper and lower solution conditions,” Boundary Value Problems, vol 2008, Article ID 197205, 21 pages, 2008... 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On the other hand, there has been recent interest in establishing fixed point theorems in partially ordered complete metric spaces with. point theorems for mixed-monotone mappings in partially ordered complete metric spaces which satisfy a weaker contraction condition than the classical Banach contraction condition for all points. continuous function satisfying the following properties: afx, y is nondecreasing in x ∈ a, b for each y ∈ a, b, and fx, y is nonincreasing in y ∈ a, b for each x ∈ a, b; 2 Fixed Point